The subject heading for this thread, 'Aristotle was Right!', refersto a longstanding debate in the literature. If you Google on'Aristotle' and 'tetrahedron', you'll find a prevalent meme: thatfor thousands of years people mistakenly believed that tetrahedrafill space, because Aristotle said so.

Indeed, twas the questioning of revered (church-certified) ancientauthorities that resulted in the Renaissance mindset, markedthe end of what, in retrospect, many came to call a "dark age"in western civilization.

Pointing out this error of Aristotle's therefore comes across asa story with a moral: questioning authority is healthy, and notdoing so may delay human progress for thousands of years.

However, if you dig more deeply into the debate, you will findthat Aristotle's apologists have often cited the fact that hedidn't say "regular" i.e. the "pyramid" to which he may havebeen referring could have been somehow irregular in shape.

This centuries-long search for space-filling tetrahedra resultedin some pioneering studies that in themselves pushed theboundaries of what we know, right down to our own times.

Majorie writes: "Aristotle did not state explicitly thathe meant regular tetrahedra... some scholars continuedto defend Aristotle on the grounds that he had notexplicitly required regularity..."

One explorer-geometer getting a lot of focus in thiswrite-up is D. M. Y. Sommerville (1879-1934) whoisolates what in contemporary nomenclature we callthe Mite, or Minimum Tetrahedron. This is depictedin Figure 10 of the Senechal monograph, as 1/24thof the cube.

Sommerville applies two important criteria to constrainhis search:

(a) the tetrahedra in question must fill space by face bonding

(b) any singular space-filler must not rely on a mirror-imageto accomplish its space-filling duties.

This Mite, in turn, face-bonds to create two other tetra-hedral space-fillers meeting Sommerville's criteria, namelythe Rite (aka a tetrahedral disphenoid) and the Bite (amono-rectangular symmetric tetrahedron), both classifiedas Sytes, i.e. those polyhedra comprised of two face-bondedMites (of which there are three, but one is a hexahedron).

So we should pause at this juncture to acknowledge thatAristotle's defenders have a strong argument: given he didnot specify "regular" then his assertion is manifestly correct.Blanket, unqualified statements to the effect that tetrahedrado not fill space are manifestly incorrect.

You'll find an example of such an incorrect statement at theMath World web site, in the entry on space-filling polyhedra:

"A space-filling polyhedron, sometimes called a a plesiohedron(Grünbaum and Shephard 1980), is a polyhedron whichcan be used to generate a tessellation of space. Althougheven Aristotle himself proclaimed in his work On the Heavensthat the tetrahedron fills space, it in fact does not."

(note also that no tetrahedra are depicted in the accompanyinggraphics, reinforcing the mis-impression given by the abovesentences).

The topic of space-filling tessellations rarely arises incontemporary K-16 mathematics, largely because spatialgeometry as a whole has been given short shrift. Evenas our technology is getting better at sharing spatialinformation, our K-16 curriculum has been getting visuallypoorer, more lexical, more algebraic, less "right brained".

Some teachers call this "flying blind on instruments" andblame the Bourbaki movement. Economic factors alsoplay a role in that textbook publishers try to get by withold figures, discourage a lot of new graphics, especiallythose requiring perspective.

Animations don't fit the textbook format at all, yet today'sstudents are brought up watching television -- resulting ina severe disconnect as animations are denied them in asubject which cries out for animated treatments. Thosemost serious about math reform at least address thisdisconnect, sometimes citing McLuhan.

Perhaps we have turned a corner and entered a newchapter in that regard, given recent advances in pedagogy.

According to our spanking new volumes chart, the Miteweighs in at 1/8, the Sytes at 1/4, relative to a cube ofvolume 3 and a regular tetrahedron of 1. These easywhole number and/or rational volumes make spatialgeometry more accessible, less intimidating. Thenewer terminology is also more memorable. We callthis a "concentric hierarchy" of polyhedra and note thelong and venerable history behind it, including Keplerand many others (a NeoPlatonist tradition).

Per earlier posts to this archive (math-teach) we traceour Mite and related modules back to the five Platonicsand the combinations of their duals.

Dissections of these shapes, by means of simple andlogical cuts, provide the derivation for our A & B particles,along with the Mite (comprised of two As and 1B). Thesebuilding blocks are suitable for elementary school use,with their derivations shown as projected screen animationsand/or accomplished with clay, paper, other materials.

Older students have access to the algebraic andtrigonometric expressions for characterizing theseobjects. The addition of vector mechanics and somecomputer programming provides a basis for our 21stcentury high school level geometry curriculum.

Chalkboard slogan:

Aristotle was RightRemember the Mite

Note: some math teachers have been discussing thepossibility of labeling the Mite "Aristotle's Tetrahedron"in his honor, and in hopes of rectifying some of thistarnishing of his reputation that has been going on forsome centuries. This may not catch on, but it's worthbringing up. You'll find more recent discussion onmathfuture (Google group) and in the blog post below.